An object 4.0 mm high is 10 cm in front of a concave mirror having a 6.0 cm focal length. Find the height by means of;
a) the lens/ mirror and magnification equations.
To find the height of the image formed by the concave mirror, we can use the lens/mirror equation and the magnification equation.
1. Lens/Mirror Equation:
The lens/mirror equation relates the object distance (do), the image distance (di), and the focal length (f) of a concave mirror. The equation is given by:
1/f = 1/do + 1/di
In this case, the object distance (do) is the distance from the object to the mirror (10 cm), and the focal length (f) of the concave mirror is given as 6.0 cm.
2. Using the Lens/Mirror Equation:
Let's start by rearranging the lens/mirror equation to solve for the image distance (di):
1/di = 1/f - 1/do
Substituting the given values:
1/di = 1/6.0 cm - 1/10.0 cm
Now, calculate the value on the right-hand side of the equation:
1/di = (10 - 6)/(6 * 10) = 4/60 = 1/15 cm
Taking the reciprocal on both sides of the equation:
di = 15 cm
So, the image distance is 15 cm.
3. Magnification Equation:
The magnification equation relates the height of the image (hi), the height of the object (ho), and the magnification (m) of the mirror. The equation is given by:
m = -di/do = hi/ho
In this case, the height of the object (ho) is given as 4.0 mm.
4. Using the Magnification Equation:
Let's rearrange the magnification equation to solve for the height of the image (hi):
hi = m * ho
The magnification (m) of a concave mirror is always negative because it produces an inverted image. Since the magnification is not given, we can assume it to be negative.
Substituting the given values:
hi = -di/do * ho = -15 cm/10 cm * 4.0 mm
Now, convert all measurements to the same units:
hi = -1.5 * 0.4 cm = -0.6 cm
So, the height of the image formed by the concave mirror is -0.6 cm. The negative sign indicates an inverted image.